63 



of algebraical geometry. He has also arrived at simplifi- 

 cations of this proof by introducing the ordinary imaginary 

 •^—1, treated by the ordinary rules; but his first investiga- 

 tion, and the one which he prefers, has been founded on the 

 rules of the calculus of quaternions, and consists in resolving 

 by those rules the system of the two equations : 



ap" + p"« _ ap -\- pa ap" — p"a _ ap — pa 

 ap' + p'a av + va ap' — p'a av — va 



in which 



a = ia -\- jb + he, 



V — iu -\-jv + kw, 



P = ix -\-jy + kz, 



p'=ix'+jij'+kz', 



p"=ix"+jy"+kz", 



i,j, k, being (as in former communications respecting quater- 

 nions) three imaginary units connected by the nine non-linear 

 relations 



i^—f—li^— - 1 ; ijzzk,jk—i, ki=J; ji=-k, kj=—i, ik=—j. 



The phrase " equations of signification" is borrowed from 

 Mr. De Morgan. If the theorem be particularized, so as to 

 correspond to that gentleman's system of triplets, by making 



ti — \, V zz tv=:0, a — — b = — c, 

 then the equations of signification reduce themselves to 



and the formula of multiplication resolves itself into the three 

 relations 



x" = xx' 4- yz' 4- zy', 



y" = xij' + yx' — zz', 



z" z=. xz' + zx' — yy' . 

 On the other hand, some of the results of the systems of the 

 two Messrs. Graves may be reproduced by taking the same 

 unit-line, u = \, v — tv — 0, but employing that other axis 

 for which a — b-zn c. The equations of signification give 

 then, more simply, 



